The Girth Ramsey Theorem

An introduction to structural Ramsey theory that builds to a resolution of one of its long-standing conjectures

Structural Ramsey theory is an area of combinatorics about Ramsey-type statements dealing with hypergraphs as well as other combinatorial structures. Rather than mainly focusing on numerical aspects such as Ramsey numbers, structural Ramsey theory focuses on the existence of Ramsey-type statements and their connection with some other fields of mathematics. In this book, Christian Reiher and Vojtech Roedl introduce structural Ramsey theory and solve a long-standing problem regarding locally sparse Ramsey graphs.

The book has two parts. The first part, which is suitable for a topics course, introduces tools used in the construction of locally sparse Ramsey graphs and hypergraphs, and covers several well-known results, such as Ramsey's theorem and the Hales-Jewett theorem. The focus is on structural Ramsey theory of hypergraphs and on the introduction of partite construction, which is illustrated by several of its applications yielding the proof of some older results in the area. The second part establishes the girth Ramsey theorem, resolving a decades' old conjecture in structural Ramsey theory implying that for every graph G and every number of colors, there exists a Ramsey graph H with girth(H) = girth(G).